A101624 -id:A101624

A260534

Square array read by ascending antidiagonals, T(n,k) = Sum_{j=0..k} n^j*(C(k-j,j) mod 2).

+0
1

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 1, 3, 1, 1, 1, 5, 1, 7, 2, 1, 1, 1, 6, 1, 13, 5, 3, 1, 1, 1, 7, 1, 21, 10, 11, 1, 1, 1, 1, 8, 1, 31, 17, 31, 1, 4, 1, 1, 1, 9, 1, 43, 26, 69, 1, 23, 3, 1, 1, 1, 10, 1, 57, 37, 131, 1, 94, 21, 5, 1, 1, 1, 11

(list; graph; refs; listen; history; text; internal format)

EXAMPLE

[2] 1, 1, 3, 1, 7, 5, 11, 1, 23, 21, 59, ... [A101624]

CROSSREFS

Cf. A002061, A002487, A002522, A002523, A059826, A071568, A101624, A101625.

A210680

a(n) = (3*a(n-1)) XOR a(n-2).

+0
1

0, 1, 3, 8, 27, 89, 272, 873, 2859, 8936, 25491, 67665, 228192, 752241, 2167859, 6833896, 18464907, 52758345, 142005584, 440498361, 1186119547, 3461957320, 9357060899, 26968655777, 72945663424, 226371206881, 613739200867, 1752444795592, 4702791627067, 14623717009785

(list; graph; refs; listen; history; text; internal format)

CROSSREFS

Cf. A101624: a(n) = (2*a(n-2)) XOR a(n-1), a(0)=0, a(1)=1.

A204771

a(n) = a(n-1) XOR (a(n-2)*3).

+0
2

0, 1, 1, 2, 1, 7, 4, 17, 29, 46, 121, 243, 408, 833, 1929, 3658, 6353, 12815, 30844, 61009, 100133, 216534, 514233, 930107, 1686288, 3352737, 8264081, 15163506, 27077825, 53153175, 133991380, 243114769, 428343405, 854649182, 2120804377, 3870970883, 6937439304

(list; graph; refs; listen; history; text; internal format)

CROSSREFS

Cf. A101624: a(n) = a(n-1) XOR (a(n-2)*2).

A101623

Modular binomial transform of 10^n.

+0
1

1, 1, 11, 1, 111, 101, 1011, 1, 10111, 10101, 111011, 10001, 1100111, 1000101, 10001011, 1, 100010111, 100010101, 1100111011, 100010001, 11101100111, 10101000101, 101110001011, 100000001, 1011000010111, 1010000010101

(list; graph; refs; listen; history; text; internal format)

COMMENTS

A101624 in binary.

A099902

Multiplies by 2 and shifts right under the XOR BINOMIAL transform (A099901).

+0
5

1, 3, 7, 11, 23, 59, 103, 139, 279, 827, 1895, 2955, 5655, 14395, 24679, 32907, 65815, 197435, 460647, 723851, 1512983, 3881019, 6774887, 9142411, 18219287, 54002491, 123733863, 192940939, 369104407, 939538491, 1610637415, 2147516555

(list; graph; refs; listen; history; text; internal format)

COMMENTS

Bisection of A101624. - Paul Barry, May 10 2005

A101625

A bisection of the Stern-Jacobsthal numbers.

+0
5

0, 1, 1, 5, 1, 21, 17, 69, 1, 277, 273, 1349, 257, 5141, 4113, 16453, 1, 65813, 65809, 329029, 65793, 1381397, 1118225, 4538437, 65537, 18088213, 17826065, 88081733, 16777473, 335549461, 268439569, 1073758277, 1, 4295033109, 4295033105

(list; graph; refs; listen; history; text; internal format)

FORMULA

a(n) = A101624(2n+1).

A006921

Diagonals of Pascal's triangle mod 2 interpreted as binary numbers.
(Formerly M2252)

+0
5

1, 1, 3, 2, 7, 5, 13, 8, 29, 21, 55, 34, 115, 81, 209, 128, 465, 337, 883, 546, 1847, 1301, 3357, 2056, 7437, 5381, 14087, 8706, 29443, 20737, 53505, 32768, 119041, 86273, 226051, 139778, 472839, 333061, 859405, 526344, 1903901, 1377557, 3606327

(list; graph; refs; listen; history; text; internal format)

CROSSREFS

Cf. A011973, A000079, A047999 (Sierpiński), A007318, A101624.

A002487

Stern's diatomic series (or Stern-Brocot sequence): a(0) = 0, a(1) = 1; for n > 0: a(2*n) = a(n), a(2*n+1) = a(n) + a(n+1).
(Formerly M0141 N0056)

+0
378

0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4, 11, 7, 10, 3, 11, 8, 13, 5, 12, 7, 9, 2, 9, 7, 12, 5, 13, 8, 11, 3, 10, 7, 11, 4, 9, 5, 6, 1, 7, 6, 11, 5, 14, 9, 13, 4, 15, 11, 18, 7, 17, 10, 13, 3, 14, 11, 19, 8, 21, 13, 18, 5, 17, 12, 19

(list; graph; refs; listen; history; text; internal format)

FORMULA

a(n) = A001222(A260443(n)) = A000120(A277020(n)). Also a(n) = A000120(A101624(n-1)) for n >= 1. - Antti Karttunen, Nov 05 2016

CROSSREFS

Cf. A000123, A000360, A001045, A002083, A011655, A020950, A026741, A037227, A046815, A070871, A070872, A071883, A073459, A084091, A101624, A126606, A174980, A174981, A178239, A178568, A212288, A213369, A260443, A277020, A277325, A287729, A287730, A293160.