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A115356 Matrix (1,x)+(x,x^2) in Riordan array notation.

%I A115356 #16 Jan 19 2025 14:16:05
%S A115356 1,1,1,0,0,1,0,1,0,1,0,0,0,0,1,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,
%T A115356 0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,
%U A115356 0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1
%N A115356 Matrix (1,x)+(x,x^2) in Riordan array notation.
%C A115356 As a flat sequence, with starting offset=1, a(n) = 1 if n is a triangular number (A000217) or twice a square (A001105), otherwise 0. - _Antti Karttunen_, Jan 19 2025
%H A115356 Antti Karttunen, <a href="/A115356/b115356.txt">Table of n, a(n) for n = 0..101474; the first 450 rows of the triangle</a>
%H A115356 Wikipedia, <a href="https://en.wikipedia.org/wiki/Riordan_array">Riordan array</a>
%H A115356 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>.
%F A115356 Number triangle T(n, k) = if(n=k, 1, 0) OR if(n=2k+1, 1, 0).
%F A115356 a(n) = A010054(n) + A379480(n). [As a flat sequence with starting offset 1] - _Antti Karttunen_, Jan 19 2025
%e A115356 Triangle begins:
%e A115356 n\k| 0  1  2  3  4  5  6  7  8  9
%e A115356 ---+-------------------------------
%e A115356 0  | 1;
%e A115356 1  | 1, 1;
%e A115356 2  | 0, 0, 1;
%e A115356 3  | 0, 1, 0, 1;
%e A115356 4  | 0, 0, 0, 0, 1;
%e A115356 5  | 0, 0, 1, 0, 0, 1;
%e A115356 6  | 0, 0, 0, 0, 0, 0, 1;
%e A115356 7  | 0, 0, 0, 1, 0, 0, 0, 1;
%e A115356 8  | 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e A115356 9  | 0, 0, 0, 0, 1, 0, 0, 0, 0, 1;
%e A115356 (row and column numbering added by _Antti Karttunen_, Jan 19 2025)
%o A115356 (PARI)
%o A115356 A115356off1(n) = (ispolygonal(n,3) || (!(n%2) && issquare(n/2))); \\ (This is one-based)
%o A115356 A115356(n) = A115356off1(1+n); \\ (zero-based) - _Antti Karttunen_, Jan 19 2025
%Y A115356 Row sums are A000034. Diagonal sums are A115357. Inverse is A115358.
%Y A115356 Cf. A000217, A001105, A010054, A379480.
%Y A115356 Cf. also A115359.
%K A115356 easy,nonn,tabl
%O A115356 0,1
%A A115356 _Paul Barry_, Jan 21 2006