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A107448
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Irregular triangle T(n, k) = b(n) + k^2 + k + 1, where b(n) = A056486(n-1) - (1/2)*[n=1], for n >= 1 and 1 <= k <= b(n) - 1, read by rows.
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3
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5, 7, 11, 17, 13, 17, 23, 31, 41, 53, 67, 83, 101, 19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, 197, 223, 251, 281, 313, 347, 383, 421, 461, 503, 547, 593, 641, 691, 743, 797, 853, 911, 971, 1033
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history;
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OFFSET
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1,1
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COMMENTS
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Former title: Triangular form sequence made from a version of A082605 Euler extension.
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REFERENCES
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Advanced Number Theory, Harvey Cohn, Dover Books, 1963, Page 155
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LINKS
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FORMULA
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T(n, k) = b(n) + k^2 + k + 1, where b(n) = A056486(n-1) - (1/2)*[n=1], for n >= 1 and 1 <= k <= b(n) - 1. - G. C. Greubel, Mar 23 2024
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EXAMPLE
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The irregular triangle begins as:
5;
7, 11, 17;
13, 17, 23, 31, 41, 53, 67, 83, 101;
19, 23, 29, 37, 47, 59, 73, 89, 107, 127, 149, 173, 199, 227, 257;
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MATHEMATICA
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(* First program *)
a[1] = 3; a[2] = 5; a[3] = 11; a[n_]:= a[n]= Abs[1-4*a[n-2]] -2;
euler= Table[a[n], {n, 10}];
Table[k^2 + k + euler[[n]], {n, 7}, {k, euler[[i]] -2}]//Flatten
(* Second program *)
b[n_]:= 2^(n-3)*(9-(-1)^n) - Boole[n==1]/2;
T[n_, k_]:= b[n] +k^2+k+1;
Table[T[n, k], {n, 8}, {k, b[n]-1}]//Flatten (* G. C. Greubel, Mar 23 2024 *)
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PROG
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(Magma)
b:= func< n | n eq 1 select 2 else 2^(n-3)*(9-(-1)^n) >;
A107448:= func< n, k | b(n) +k^2 +k +1 >;
(SageMath)
def b(n): return 2^(n-3)*(9-(-1)^n) - int(n==1)/2
def A107448(n, k): return b(n) + k^2+k+1;
flatten([[A107448(n, k) for k in range(1, b(n))] for n in range(1, 8)]) # G. C. Greubel, Mar 23 2024
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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